Section 5 EXPRESSIONS AND EQUATIONS

In the section you will learn:

ü o expressions and their simplifications;

ü what are the properties of equalities;

ü how to solve equations based on the properties of equalities;

ü what types of problems are solved with the help of equations; what are perpendicular lines and how to build them;

ü what lines are called parallel and how to build them;

ü what is a coordinate plane;

ü how to determine the coordinates of a point on a plane;

ü what is a dependency graph between quantities and how to build it;

ü how to apply the learned material in practice

§ 30. EXPRESSIONS AND THEIR SIMPLIFICATION

You already know what literal expressions are and know how to simplify them using the laws of addition and multiplication. For example, 2a ∙ (-4 b) = -8 ab . In the resulting expression, the number -8 is called the coefficient of the expression.

Does the expression cd coefficient? So. It is equal to 1 because cd - 1 ∙ cd .

Recall that converting an expression with parentheses to an expression without parentheses is called parenthesis expansion. For example: 5(2x + 4) = 10x + 20.

The reverse action in this example is to put the common factor out of brackets.

Terms containing the same literal factors are called similar terms. By taking the common factor out of brackets, similar terms are erected:

5x + y + 4 - 2x + 6 y - 9 =

= (5x - 2x) + (y + 6y )+ (4 - 9) = = (5-2)* + (1 + 6)* y-5=

B x + 7y - 5.

Bracket expansion rules

1. If there is a “+” sign in front of the brackets, then when opening the brackets, the signs of the terms in brackets are preserved;

2. If there is a “-” sign in front of the brackets, then when the brackets are opened, the signs of the terms in brackets are reversed.

Task 1 . Simplify the expression:

1) 4x+(-7x + 5);

2) 15 y -(-8 + 7 y ).

Solutions. 1. There is a “+” sign before the brackets, therefore, when opening the brackets, the signs of all terms are preserved:

4x + (-7x + 5) \u003d 4x - 7x + 5 \u003d -3x + 5.

2. There is a “-” sign in front of the brackets, therefore, during the opening of the brackets: the signs of all terms are reversed:

15 - (- 8 + 7y) \u003d 15y + 8 - 7y \u003d 8y +8.

To open brackets, use the distributive property of multiplication: a( b + c) = ab + ac. If a > 0, then the signs of the terms b and with do not change. If a< 0, то знаки слагаемых b and from are reversed.

Task 2. Simplify the expression:

1) 2(6y -8) + 7y;

2) -5 (2-5x) + 12.

Solutions. 1. The factor 2 in front of the brackets e is positive, therefore, when opening the brackets, we keep the signs of all terms: 2(6 y - 8) + 7 y = 12 y - 16 + 7 y =19 y -16.

2. The factor -5 in front of the brackets e is negative, therefore, when opening the brackets, we change the signs of all terms to the opposite ones:

5(2 - 5x) + 12 = -10 + 25x +12 = 2 + 25x.

Find out more

1. The word "sum" comes from the Latin summa , which means "total", "total".

2. The word "plus" comes from the Latin plus , which means "more", and the word "minus" - from the Latin minus , which means "less". The signs "+" and "-" are used to indicate the operations of addition and subtraction. These signs were introduced by the Czech scientist J. Vidman in 1489 in the book "A quick and pleasant account for all merchants"(Fig. 138).

Rice. 138

REMEMBER THE MAIN THINGS

1. What terms are called similar? How are like terms constructed?

2. How do you open brackets preceded by a “+” sign?

3. How do you open brackets preceded by a "-" sign?

4. How do you open brackets that are preceded by a positive factor?

5. How do you open brackets that are preceded by a negative factor?

1374". Name the coefficient of the expression:

1) 12 a; 3) -5.6 xy;

2)4 6; 4)-s.

1375". Name the terms that differ only by the coefficient:

1) 10a + 76-26 + a; 3) 5n + 5m -4n + 4;

2) bc -4d - bc + 4d; 4) 5x + 4y-x + y.

What are these terms called?

1376". Are there similar terms in the expression:

1) 11a + 10a; 3)6n + 15n; 5) 25r - 10r + 15r;

2) 14s-12; 4)12 m + m; 6) 8k +10k - n?

1377". Is it necessary to change the signs of the terms in brackets, opening the brackets in the expression:

1)4 + (a + 3b); 2)-c +(5-d ); 3) 16-(5m-8n)?

1378°. Simplify the expression and underline the coefficient:

1379°. Simplify the expression and underline the coefficient:

1380°. Reduce like terms:

1) 4a - Po + 6a - 2a; 4) 10 - 4 d - 12 + 4d;

2) 4b - 5b + 4 + 5b; 5) 5a - 12b - 7a + 5b;

3)-7ang="EN-US">c+ 5-3 c + 2; 6) 14 n - 12 m -4 n -3 m.

1381°. Reduce like terms:

1) 6a - 5a + 8a -7a; 3) 5s + 4-2s-3s;

2)9 b +12-8-46; 4) -7n + 8m - 13n - 3m.

1382°. Take the common factor out of brackets:

1) 1.2 a +1.2 b; 3) -3 n - 1.8 m; 5) -5p + 2.5k -0.5t;

2) 0.5 s + 5d; 4) 1.2 n - 1.8 m; 6) -8p - 10k - 6t.

1383°. Take the common factor out of brackets:

1) 6a-12b; 3) -1.8 n -3.6 m;

2) -0.2 s + 1 4 d; A) 3p - 0.9k + 2.7t.

1384°. Open brackets and reduce like terms;

1) 5 + (4a -4); 4) -(5 c - d) + (4 d + 5c);

2) 17x-(4x-5); 5) (n - m) - (-2 m - 3 n);

3) (76 - 4) - (46 + 2); 6) 7 (-5x + y) - (-2y + 4x) + (x - 3y).

1385°. Open the brackets and reduce like terms:

1) 10a + (4 - 4a); 3) (s - 5 d) - (- d + 5s);

2) -(46-10) + (4-56); 4) - (5 n + m) + (-4 n + 8 m) - (2 m -5 n).

1386°. Expand the brackets and find the meaning of the expression:

1)15+(-12+ 4,5); 3) (14,2-5)-(12,2-5);

2) 23-(5,3-4,7); 4) (-2,8 + 13)-(-5,6 + 2,8) + (2,8-13).

1387°. Expand the brackets and find the meaning of the expression:

1) (14- 15,8)- (5,8 + 4);

2)-(18+22,2)+ (-12+ 22,2)-(5- 12).

1388°. Open parenthesis:

1) 0.5 ∙ (a + 4); 4) (n - m) ∙ (-2.4 p);

2)-s ∙ (2.7-1.2 d ); 5) 3 ∙ (-1.5 p + k - 0.2 t);

3) 1.6 ∙ (2n + m); 6) (4.2 p - 3.5 k -6 t) ∙ (-2a).

1389°. Open parenthesis:

1) 2.2 ∙ (x-4); 3)(4 c - d )∙(-0.5 y );

2) -2 ∙ (1.2 n - m); 4) 6- (-p + 0.3 k - 1.2 t).

1390. Simplify the expression:

1391. Simplify the expression:

1392. Reduce like terms:

1393. Reduce like terms:

1394. Simplify the expression:

1) 2.8 - (0.5 a + 4) - 2.5 ∙ (2a - 6);

2) -12 ∙ (8 - 2, by) + 4.5 ∙ (-6 y - 3.2);

4) (-12.8 m + 24.8 n) ∙ (-0.5)-(3.5 m -4.05 m) ∙ 2.

1395. Simplify the expression:

1396. Find the meaning of the expression;

1) 4-(0.2 a-3) - (5.8 a-16), if a \u003d -5;

2) 2-(7-56)+ 156-3∙(26+ 5), if = -0.8;

m = 0.25, n = 5.7.

1397. Find the value of the expression:

1) -4∙ (i-2) + 2∙(6x - 1), if x = -0.25;

1398*. Find the error in the solution:

1) 5- (a-2.4) -7 ∙ (-a + 1.2) \u003d 5a - 12-7a + 8.4 \u003d -2a-3.6;

2) -4 ∙ (2.3 a - 6) + 4.2 ∙ (-6 - 3.5 a) \u003d -9.2 a + 46 + 4.26 - 14.7 a \u003d -5.5 a + 8.26.

1399*. Expand the brackets and simplify the expression:

1) 2ab - 3(6(4a - 1) - 6(6 - 10a)) + 76;

1400*. Arrange the parentheses to get the correct equality:

1) a-6-a + 6 \u003d 2a; 2) a -2 b -2 a + b \u003d 3 a -3 b.

1401*. Prove that for any numbers a and b if a > b , then the following equality holds:

1) (a + b) + (a-b) \u003d 2a; 2) (a + b) - (a - b) \u003d 2 b.

Will this equality be correct if: a) a< b; b) a = 6?

1402*. Prove that for any natural number a, the arithmetic mean of the preceding and following numbers is equal to a.

APPLY IN PRACTICE

1403. To prepare a fruit dessert for three people, you need: 2 apples, 1 orange, 2 bananas and 1 kiwi. How to make a literal expression to determine the amount of fruit needed to prepare a dessert for guests? Help Marin to calculate how many fruits she needs to buy if she comes to visit: 1) 5 friends; 2) 8 friends.

1404. Make a literal expression to determine the time required to complete homework in mathematics, if:

1) a min was spent on solving problems; 2) simplification of expressions is 2 times more than for solving problems. How much time did Vasilko do his homework if he spent 15 minutes solving problems?

1405. Lunch in the school canteen consists of salad, borscht, cabbage rolls and compote. The cost of salad is 20%, borscht - 30%, cabbage rolls - 45%, compote - 5% of the total cost of the entire meal. Write an expression to find the cost of lunch at the school cafeteria. How much does lunch cost if the price of a salad is 2 UAH?

REPETITION TASKS

1406. Solve the equation:

1407. Tanya spent on ice creamall available money, and for sweets -the rest. How much money does Tanya have?

if sweets cost 12 UAH?

Some algebraic examples of one kind are capable of terrifying schoolchildren. Long expressions are not only intimidating, but also very difficult to calculate. Trying to immediately understand what follows and what follows, not to get confused for long. It is for this reason that mathematicians always try to simplify the “terrible” task as much as possible and only then proceed to solve it. Oddly enough, such a trick greatly speeds up the process.

Simplification is one of the fundamental points in algebra. If in simple tasks it is still possible to do without it, then more difficult to calculate examples may be “too tough”. This is where these skills come in handy! Moreover, complex mathematical knowledge is not required: it will be enough just to remember and learn how to put into practice a few basic techniques and formulas.

Regardless of the complexity of the calculations, when solving any expression, it is important follow the order of operations with numbers:

1. parentheses;
2. exponentiation;
3. multiplication;
4. division;
5. addition;
6. subtraction.

The last two points can be safely swapped and this will not affect the result in any way. But adding two neighboring numbers, when next to one of them there is a multiplication sign, is absolutely impossible! The answer, if any, is wrong. Therefore, you need to remember the sequence.

The use of such

Such elements include numbers with a variable of the same order or the same degree. There are also so-called free members that do not have next to them the letter designation of the unknown.

The bottom line is that in the absence of parentheses You can simplify the expression by adding or subtracting like.

A few illustrative examples:

• 8x 2 and 3x 2 - both numbers have the same second order variable, so they are similar and when added, they are simplified to (8+3)x 2 =11x 2, while when subtracted, it turns out (8-3)x 2 =5x 2;
• 4x 3 and 6x - and here "x" has a different degree;
• 2y 7 and 33x 7 - contain different variables, therefore, as in the previous case, they do not belong to similar ones.

Factoring a Number

This little mathematical trick, if you learn how to use it correctly, will help you to cope with a tricky problem more than once in the future. And it’s easy to understand how the “system” works: a decomposition is a product of several elements, the calculation of which gives the original value. Thus, 20 can be represented as 20x1, 2x10, 5x4, 2x5x2, or some other way.

On a note: multipliers are always the same as divisors. So you need to look for a working “pair” for expansion among the numbers by which the original is divisible without a remainder.

You can perform such an operation both with free members and with digits attached to a variable. The main thing is not to lose the latter during calculations - even after decomposition, the unknown cannot take and "go nowhere." It remains at one of the factors:

• 15x=3(5x);
• 60y 2 \u003d (15y 2) 4.

Prime numbers that can only be divided by themselves or 1 never factor - it makes no sense..

Basic Simplification Methods

The first thing that catches the eye:

• the presence of brackets;
• fractions;
• roots.

Algebraic examples in the school curriculum are often compiled with the assumption that they can be beautifully simplified.

Bracket Calculations

Pay close attention to the sign in front of the brackets! Multiplication or division is applied to each element inside, and minus - reverses the existing "+" or "-" signs.

Parentheses are calculated according to the rules or according to the formulas of abbreviated multiplication, after which similar ones are given.

Fraction reduction

Reduce fractions is also easy. They themselves “willingly run away” once in a while, it is worth making operations with bringing such members. But you can simplify the example even before this: pay attention to the numerator and denominator. They often contain explicit or hidden elements that can be mutually reduced. True, if in the first case you just need to delete the superfluous, in the second you will have to think, bringing part of the expression to the form for simplification. Methods used:

• search and bracketing of the greatest common divisor of the numerator and denominator;
• dividing each top element by the denominator.

When an expression or part of it is under the root, the primary simplification problem is almost the same as the case with fractions. It is necessary to look for ways to completely get rid of it or, if this is not possible, to minimize the sign interfering with calculations. For example, to unobtrusive √(3) or √(7).

A sure way to simplify the radical expression is to try to factor it out, some of which are outside the sign. An illustrative example: √(90)=√(9×10) =√(9)×√(10)=3√(10).

Other little tricks and nuances:

• this simplification operation can be carried out with fractions, taking it out of the sign both as a whole and separately as a numerator or denominator;
• it is impossible to decompose and take out a part of the sum or difference beyond the root;
• when working with variables, be sure to take into account its degree, it must be equal to or a multiple of the root for the possibility of rendering: √(x 2 y)=x√(y), √(x 3)=√(x 2 ×x)=x√( x);
• sometimes it is allowed to get rid of the radical variable by raising it to a fractional power: √ (y 3)=y 3/2.

Power Expression Simplification

If in the case of simple calculations by minus or plus, examples are simplified by bringing similar ones, then what about when multiplying or dividing variables with different powers? They can be easily simplified by remembering two main points:

1. If there is a multiplication sign between the variables, the exponents are added.
2. When they are divided by each other, the same denominator is subtracted from the degree of the numerator.

The only condition for such a simplification is that both terms have the same basis. Examples for clarity:

• 5x 2 × 4x 7 + (y 13 / y 11) \u003d (5 × 4)x 2+7 + y 13- 11 \u003d 20x 9 + y 2;
• 2z 3 +z×z 2 -(3×z 8 /z 5)=2z 3 +z 1+2 -(3×z 8-5)=2z 3 +z 3 -3z 3 =3z 3 -3z 3 = 0.

We note that operations with numerical values ​​in front of variables occur according to the usual mathematical rules. And if you look closely, it becomes clear that the power elements of the expression "work" in a similar way:

• raising a member to a power means multiplying it by itself a certain number of times, i.e. x 2 \u003d x × x;
• division is similar: if you expand the degree of the numerator and denominator, then some of the variables will be reduced, while the rest are “gathered”, which is equivalent to subtraction.

As in any business, when simplifying algebraic expressions, not only knowledge of the basics is necessary, but also practice. After just a few lessons, examples that once seemed complicated will be reduced without much difficulty, turning into short and easily solved ones.

Video

This video will help you understand and remember how expressions are simplified.

Didn't get an answer to your question? Suggest a topic to the authors.

Among the various expressions that are considered in algebra, sums of monomials occupy an important place. Here are examples of such expressions:
\(5a^4 - 2a^3 + 0.3a^2 - 4.6a + 8 \)
\(xy^3 - 5x^2y + 9x^3 - 7y^2 + 6x + 5y - 2 \)

The sum of monomials is called a polynomial. The terms in a polynomial are called members of the polynomial. Mononomials are also referred to as polynomials, considering a monomial as a polynomial consisting of one member.

For example, polynomial
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 \)
can be simplified.

We represent all the terms as monomials of the standard form:
\(8b^5 - 2b \cdot 7b^4 + 3b^2 - 8b + 0.25b \cdot (-12)b + 16 = \)
\(= 8b^5 - 14b^5 + 3b^2 -8b -3b^2 + 16 \)

We give similar terms in the resulting polynomial:
\(8b^5 -14b^5 +3b^2 -8b -3b^2 + 16 = -6b^5 -8b + 16 \)
The result is a polynomial, all members of which are monomials of the standard form, and among them there are no similar ones. Such polynomials are called polynomials of standard form.

Behind polynomial degree standard form take the largest of the powers of its members. So, the binomial \(12a^2b - 7b \) has the third degree, and the trinomial \(2b^2 -7b + 6 \) has the second.

Usually, the terms of standard form polynomials containing one variable are arranged in descending order of its exponents. For example:
\(5x - 18x^3 + 1 + x^5 = x^5 - 18x^3 + 5x + 1 \)

The sum of several polynomials can be converted (simplified) into a standard form polynomial.

Sometimes the members of a polynomial need to be divided into groups, enclosing each group in parentheses. Since parentheses are the opposite of parentheses, it is easy to formulate parentheses opening rules:

If the + sign is placed before the brackets, then the terms enclosed in brackets are written with the same signs.

If a "-" sign is placed in front of the brackets, then the terms enclosed in brackets are written with opposite signs.

Transformation (simplification) of the product of a monomial and a polynomial

Using the distributive property of multiplication, one can transform (simplify) the product of a monomial and a polynomial into a polynomial. For example:
\(9a^2b(7a^2 - 5ab - 4b^2) = \)
\(= 9a^2b \cdot 7a^2 + 9a^2b \cdot (-5ab) + 9a^2b \cdot (-4b^2) = \)
\(= 63a^4b - 45a^3b^2 - 36a^2b^3 \)

The product of a monomial and a polynomial is identically equal to the sum of the products of this monomial and each of the terms of the polynomial.

This result is usually formulated as a rule.

To multiply a monomial by a polynomial, one must multiply this monomial by each of the terms of the polynomial.

We have repeatedly used this rule for multiplying by a sum.

The product of polynomials. Transformation (simplification) of the product of two polynomials

In general, the product of two polynomials is identically equal to the sum of the product of each term of one polynomial and each term of the other.

Usually use the following rule.

To multiply a polynomial by a polynomial, you need to multiply each term of one polynomial by each term of the other and add the resulting products.

Abbreviated multiplication formulas. Sum, Difference, and Difference Squares

Some expressions in algebraic transformations have to be dealt with more often than others. Perhaps the most common expressions are \((a + b)^2, \; (a - b)^2 \) and \(a^2 - b^2 \), that is, the square of the sum, the square of the difference, and square difference. You noticed that the names of the indicated expressions seem to be incomplete, so, for example, \((a + b)^2 \) is, of course, not just the square of the sum, but the square of the sum of a and b. However, the square of the sum of a and b is not so common, as a rule, instead of the letters a and b, it contains various, sometimes quite complex expressions.

Expressions \((a + b)^2, \; (a - b)^2 \) are easy to convert (simplify) into polynomials of the standard form, in fact, you have already met with such a task when multiplying polynomials:
\((a + b)^2 = (a + b)(a + b) = a^2 + ab + ba + b^2 = \)
\(= a^2 + 2ab + b^2 \)

The resulting identities are useful to remember and apply without intermediate calculations. Short verbal formulations help this.

\((a + b)^2 = a^2 + b^2 + 2ab \) - the square of the sum is equal to the sum of the squares and the double product.

\((a - b)^2 = a^2 + b^2 - 2ab \) - the square of the difference is the sum of the squares without doubling the product.

\(a^2 - b^2 = (a - b)(a + b) \) - the difference of squares is equal to the product of the difference and the sum.

These three identities allow in transformations to replace their left parts with right ones and vice versa - right parts with left ones. The most difficult thing in this case is to see the corresponding expressions and understand what the variables a and b are replaced in them. Let's look at a few examples of using abbreviated multiplication formulas.

In the fifth century BC, the ancient Greek philosopher Zeno of Elea formulated his famous aporias, the most famous of which is the aporia "Achilles and the tortoise". Here's how it sounds:

Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.

This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.

From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, this looks like a slowdown in time until it stops completely at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.

If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."

How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:

In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. During the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl one hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.

This approach adequately describes reality without any logical paradoxes. But this is not a complete solution to the problem. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.

Another interesting aporia of Zeno tells of a flying arrow:

A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.

In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.

Wednesday, July 4, 2018

Very well the differences between set and multiset are described in Wikipedia. We look.

As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.

Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.

No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that inextricably connects them with reality. This umbilical cord is money. Let us apply mathematical set theory to mathematicians themselves.

We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the whole amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.

First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will frantically recall physics: different coins have different amounts of dirt, the crystal structure and arrangement of atoms for each coin is unique ...

And now I have the most interesting question: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.

Look here. We select football stadiums with the same field area. The area of ​​the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.

To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."

Sunday, March 18, 2018

The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.

Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.

Let's figure out what and how we do in order to find the sum of the digits of a given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.

1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.

2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.

3. Convert individual graphic characters to numbers. This is not a mathematical operation.

4. Add up the resulting numbers. Now that's mathematics.

The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that is not all.

From the point of view of mathematics, it does not matter in which number system we write the number. So, in different number systems, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.

As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of ​​a rectangle in meters and centimeters would give you completely different results.

Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.

The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.

What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.

Sign on the door Opens the door and says:

Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?

Female... A halo on top and an arrow down is male.

If you have such a work of design art flashing before your eyes several times a day,

Then it is not surprising that you suddenly find a strange icon in your car:

Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I do not consider this girl a fool who does not know physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.

1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.

Simplifying algebraic expressions is one of the keys to learning algebra and an extremely useful skill for all mathematicians. Simplification allows you to reduce a complex or long expression to a simple expression that is easy to work with. Basic simplification skills are good even for those who are not enthusiastic about mathematics. By following a few simple rules, many of the most common types of algebraic expressions can be simplified without any special mathematical knowledge.

Steps

Important Definitions

1. Similar members. These are members with a variable of the same order, members with the same variables, or free members (members that do not contain a variable). In other words, like terms include one variable to the same extent, include several identical variables, or do not include a variable at all. The order of the terms in the expression does not matter.

   • For example, 3x 2 and 4x 2 are like terms because they contain the variable "x" of the second order (in the second power). However, x and x 2 are not similar members, since they contain the variable "x" of different orders (first and second). Similarly, -3yx and 5xz are not similar members because they contain different variables.

2. Factorization. This is finding such numbers, the product of which leads to the original number. Any original number can have several factors. For example, the number 12 can be decomposed into the following series of factors: 1 × 12, 2 × 6 and 3 × 4, so we can say that the numbers 1, 2, 3, 4, 6 and 12 are factors of the number 12. The factors are the same as divisors , that is, the numbers by which the original number is divisible.

   • For example, if you want to factor the number 20, write it like this: 4×5.
   • Note that when factoring, the variable is taken into account. For example, 20x = 4(5x).
   • Prime numbers cannot be factored because they are only divisible by themselves and 1.

3. Remember and follow the order of operations to avoid mistakes.

   • Parentheses
   • Degree
   • Multiplication
   • Division
   • Addition
   • Subtraction

   Casting Like Members

   1. Write down the expression. The simplest algebraic expressions (which do not contain fractions, roots, and so on) can be solved (simplified) in just a few steps.

      • For example, simplify the expression 1 + 2x - 3 + 4x.

   2. Define similar members (members with a variable of the same order, members with the same variables, or free members).

      • Find similar terms in this expression. The terms 2x and 4x contain a variable of the same order (first). Also, 1 and -3 are free members (do not contain a variable). Thus, in this expression, the terms 2x and 4x are similar, and the members 1 and -3 are also similar.

   3. Give similar members. This means adding or subtracting them and simplifying the expression.

      • 2x+4x= 6x
      • 1 - 3 = -2

   4. Rewrite the expression taking into account the given members. You will get a simple expression with fewer terms. The new expression is equal to the original.

      • In our example: 1 + 2x - 3 + 4x = 6x - 2, that is, the original expression is simplified and easier to work with.

   5. Observe the order in which operations are performed when casting like terms. In our example, it was easy to bring similar terms. However, in the case of complex expressions in which members are enclosed in brackets and fractions and roots are present, it is not so easy to bring such terms. In these cases, follow the order of operations.

      • For example, consider the expression 5(3x - 1) + x((2x)/(2)) + 8 - 3x. Here it would be a mistake to immediately define 3x and 2x as like terms and quote them, because the parentheses need to be expanded first. Therefore, perform the operations in their order.
        • 5(3x-1) + x((2x)/(2)) + 8 - 3x
        • 15x - 5 + x(x) + 8 - 3x
        • 15x - 5 + x 2 + 8 - 3x. Now, when the expression contains only addition and subtraction operations, you can cast like terms.
        • x 2 + (15x - 3x) + (8 - 5)
        • x 2 + 12x + 3

   Parenthesizing the multiplier

   1. Find the greatest common divisor (gcd) of all coefficients of the expression. GCD is the largest number by which all coefficients of the expression are divisible.

      • For example, consider the equation 9x 2 + 27x - 3. In this case, gcd=3, since any coefficient of this expression is divisible by 3.

   2. Divide each term of the expression by gcd. The resulting terms will contain smaller coefficients than in the original expression.

      • In our example, divide each expression term by 3.
        • 9x2/3=3x2
        • 27x/3=9x
        • -3/3 = -1
        • It turned out the expression 3x2 + 9x-1. It is not equal to the original expression.

   3. Write the original expression as equal to the product of gcd times the resulting expression. That is, enclose the resulting expression in brackets, and put the GCD out of brackets.

      • In our example: 9x 2 + 27x - 3 = 3(3x 2 + 9x - 1)

   4. Simplifying fractional expressions by taking the multiplier out of brackets. Why just take the multiplier out of brackets, as was done earlier? Then, to learn how to simplify complex expressions, such as fractional expressions. In this case, putting the factor out of the brackets can help get rid of the fraction (from the denominator).

      • For example, consider the fractional expression (9x 2 + 27x - 3)/3. Use parentheses to simplify this expression.
        • Factor out the factor 3 (as you did before): (3(3x 2 + 9x - 1))/3
        • Note that both the numerator and denominator now have the number 3. This can be reduced, and you get the expression: (3x 2 + 9x - 1) / 1
        • Since any fraction that has the number 1 in the denominator is just equal to the numerator, the original fractional expression is simplified to: 3x2 + 9x-1.

   Additional Simplification Techniques

4. Consider a simple example: √(90). The number 90 can be decomposed into the following factors: 9 and 10, and from 9 take the square root (3) and take 3 out from under the root.
   • √(90)
   • √(9×10)
   • √(9)×√(10)
   • 3×√(10)
   • 3√(10)

5. Simplifying expressions with powers. In some expressions, there are operations of multiplication or division of terms with a degree. In the case of multiplication of terms with one base, their degrees are added; in the case of dividing terms with the same base, their degrees are subtracted.

   • For example, consider the expression 6x 3 × 8x 4 + (x 17 / x 15). In the case of multiplication, add the exponents, and in the case of division, subtract them.
     • 6x 3 × 8x 4 + (x 17 / x 15)
     • (6 × 8)x 3 + 4 + (x 17 - 15)
     • 48x7+x2
   • The following is an explanation of the rule for multiplying and dividing terms with a degree.
     • Multiplying terms with powers is equivalent to multiplying terms by themselves. For example, since x 3 = x × x × x and x 5 = x × x × x × x × x, then x 3 × x 5 = (x × x × x) × (x × x × x × x × x), or x 8 .
     • Similarly, dividing terms with powers is equivalent to dividing terms by themselves. x 5 /x 3 \u003d (x × x × x × x × x) / (x × x × x). Since similar terms that are in both the numerator and the denominator can be reduced, the product of two "x", or x 2, remains in the numerator.

• Always be aware of the signs (plus or minus) in front of the terms of an expression, as many people have difficulty choosing the right sign.
• Ask for help if needed!
• Simplifying algebraic expressions is not easy, but if you get your hands on it, you can use this skill for a lifetime.